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III.47 L-Functions

Kevin Buzzard

1 How Can We “Package” a Sequence of Numbers?

Suppose we are given a sequence of numbers such as

How can we package up this sequence into one object that remembers everything about the sequence, and that might even give us new insights into the sequence? One standard technique is to use a GENERATING FUNCTION [III.32], but here is another way, which has proved very fruitful in number theory and elsewhere. Given a sequence a1, a2, a3, . . . , we define the Dirichlet series

Here, s could be a positive integer, ...

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